4'5 Integration by Substitution

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4.5 Integration by Substitution

• "Millions saw the apple fall, but Newton asked
  why." - Bernard Baruch

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Objective

• To integrate by using u-substitution

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(No Transcript)
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2 ways

• Pattern recognition
• Change in variables
• Both use u-substitution one mentally and one
  written out

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Pattern recognition

• Chain rule
• Antiderivative of chain rule

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Antidifferentiation of a Composite Function

• Let g be a function whose range is an interval I
  and let f be a function that is continuous on I.
  If g is differentiable on its domain and F is an
  antiderivative of f on I then

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In other words.

• Integral of f(u)du where u is the inside function
  and du is the derivative of the inside function

If u g(x), then du g(x)dx and
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Recognizing patterns
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What about
What is our constant multiple rule?
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REMEMBER

• The contant multiple rule only applies to
  constants!!!!!
• You CANNOT multiply and divide by a variable and
  then move the variable outside the integral sign.
  For instance

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In summary

• Pattern recognition
• Look for inside and outside functions in integral
• Determine what u and du would be
• Take integral
• Check by taking the derivative!

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Change of Variables
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Another example
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A third example
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Guidelines for making a change of variables

• 1. Choose a u g(x)
• 2. Compute du
• 3. Rewrite the integral in terms of u
• 4. Evalute the integral in terms of u
• 5. Replace u by g(x)
• 6. Check your answer by differentiating

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Try
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The General Power rule for Integration

• If g is a differentiable function of x, then
• Equivalently, if u g(x), then

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Change of variables for definite integrals

• Thm If the function u g(x) has a continuous
  derivative on the closed interval a,b and f is
  continuous on the range of g, then

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First way
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Second way
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Another example (way 1)
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Way 2
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Even and Odd functions

• Let f be integrable on the closed interval
  -a,a
• If f is an even function, then
• If f is an odd function then